NAG Library Routine Document

C06EKF

if $JOB = 1$ , the discrete convolution of $x$ and $y$ , defined by
$z_{k} = \sum_{j = 0}^{n - 1} x_{j} y_{k - j} = \sum_{j = 0}^{n - 1} x_{k - j} y_{j};$
if $JOB = 2$ , the discrete correlation of $x$ and $y$ defined by
$w_{k} = \sum_{j = 0}^{n - 1} x_{j} y_{k + j} .$

Here

x

and

y

are real vectors, assumed to be periodic, with period

n

, i.e.,

x_{j} = x_{j \pm n} = x_{j \pm 2 n} = \dots

;

z

and

w

are then also periodic with period

n

Note: this usage of the terms ‘convolution’ and ‘correlation’ is taken from Brigham (1974). The term ‘convolution’ is sometimes used to denote both these computations.

\hat{x}

\hat{y}

\hat{z}

and

\hat{w}

are the discrete Fourier transforms of these sequences, i.e.,

{\hat{x}}_{k} = \frac{1}{\sqrt{n}} \sum_{j = 0}^{n - 1} x_{j} \times \exp (- i \frac{2 π j k}{n}), etc.,

then

{\hat{z}}_{k} = \sqrt{n} . {\hat{x}}_{k} {\hat{y}}_{k}

and

{\hat{w}}_{k} = \sqrt{n} . {\bar{\hat{x}}}_{k} {\hat{y}}_{k}

(the bar denoting complex conjugate).

This routine calls the same auxiliary routines as C06EAF and C06EBF to compute discrete Fourier transforms, and there are some restrictions on the value of

n

4 References

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall

5 Parameters

1: JOB – INTEGERInput

On entry: the computation to be performed.

$JOB = 1$: $z_{k} = \sum_{j = 0}^{n - 1} x_{j} y_{k - j}$ (convolution);
$JOB = 2$: $w_{k} = \sum_{j = 0}^{n - 1} x_{j} y_{k + j}$ (correlation).

Constraint:

JOB = 1

2

2: X(N) – REAL (KIND=nag_wp) arrayInput/Output

On entry: the elements of one period of the vector

x

. If X is declared with bounds

(0 : N - 1)

in the subroutine from which C06EKF is called, then

X (j)

must contain

x_{j}

, for

j = 0, 1, \dots, n - 1

On exit: the corresponding elements of the discrete convolution or correlation.

3: Y(N) – REAL (KIND=nag_wp) arrayInput/Output

On entry: the elements of one period of the vector

y

. If Y is declared with bounds

(0 : N - 1)

in the subroutine from which C06EKF is called, then

Y (j)

must contain

y_{j}

, for

j = 0, 1, \dots, n - 1

On exit: the discrete Fourier transform of the convolution or correlation returned in the array X; the transform is stored in Hermitian form. If the components of the transform are:

\begin{array}{l} {\hat{Z}}_{k} = a_{k} + i b_{k} \\ {\hat{Z}}_{n - k} = a_{k} - i b_{k} \end{array}\} k = 0, 1, \dots n / 2

where

b_{0}

and

b_{n / 2}

when

n

is even then

X (k + 1)

holds

a_{k}

and

X (n - k + 1)

holds nonzero

b_{k}

(see Section 2.1.2 in the C06 Chapter Introduction).

4: N – INTEGERInput

On entry:

n

, the number of values in one period of the vectors X and Y. The largest prime factor of N must not exceed

19

, and the total number of prime factors of N, counting repetitions, must not exceed

20

Constraint:

N > 1

5: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: At least one of the prime factors of N is greater than $19$ .

$IFAIL = 2$: N has more than $20$ prime factors.

$IFAIL = 3$

On entry,

N \leq 1

$IFAIL = 4$

On entry,

JOB \neq 1

2

7 Accuracy

The results should be accurate to within a small multiple of the machine precision.

8 Further Comments

The time taken is approximately proportional to

n \times \log n

, but also depends on the factorization of

n

. C06EKF is faster if the only prime factors of

n

are

2

3

5

; and fastest of all if

n

is a power of

2

On the other hand, C06EKF is particularly slow if

n

has several unpaired prime factors, i.e., if the ‘square-free’ part of

n

has several factors. For such values of

n

, C06FKF (which requires additional real workspace) is considerably faster.

9 Example

This example reads in the elements of one period of two real vectors

x

and

y

, and prints their discrete convolution and correlation (as computed by C06EKF). In realistic computations the number of data values would be much larger.

NAG Library Routine DocumentC06EKF

+− Contents

1 Purpose

2 Specification

3 Description